First-passage percolation on width-two stretches with exponential link weights

We consider the first-passage percolation problem on effectively one-dimensional graphs with vertex set {1,...,n}\times{0,1} and translation-invariant edge-structure. For three of six non-trivial cases we obtain exact expressions for the asymptotic percolation rate \chi\ by solving certain recursive distributional equations and invoking results from ergodic theory to identify \chi\ as the expected asymptotic one-step growth of the first-passage time from (0,0) to (n,0).